My only concern is stuff that sticks to the tire and being thrown at the brake like constantly like grass, mud, stuff like that. I don't think it would be an issue. Even things getting rapped around the wheel where the caliper and disk are.

Where the heck are you mowing? Brakes work fine on my 4 wheeler in the mud in the creek and in tall grass. 25 years ago we did not have seat belts in cars. Pleople used to say just drive careful. To the person that wants brakes on back of mower there is no need for that. Just put them on the front and it will stop just fine.